# Mean-Variance Analysis

## What is a 'Mean-Variance Analysis'

A mean-variance analysis is the process of weighing risk (variance) against expected return. By looking at the expected return and variance of an asset, investors attempt to make more efficient investment choices â€“ seeking the lowest variance for a given expected return or seeking the highest expected return for a given variance level.

## BREAKING DOWN 'Mean-Variance Analysis'

Mean-variance analysis is a component of modern portfolio theory, which assumes investors make rational decisions and expect a higher return for increased risk. There are two major factors in mean-variance analysis: variance and expected return. Variance represents how spread-out the data set numbers are, such as the variability in daily or weekly returns of an individual security. The expected return is a subjective probability assessment on the return of the stock. If two investments have the same expected return, but one has a lower variance, the one with the lower variance is the better choice.

Different levels of diversification can be achieved in a portfolio by combining stocks with different variances and expected returns.

## Example Calculations

A portfolio's expected return is the sum of each component security's expected return multiplied by its weight in the portfolio. For example, assume the following two investments are in a portfolio:

Investment A: value = \$100,000 and expected return of 5%

Investment B: value = \$300,000 and expected return of 10%

Considering a total portfolio value of \$400,000, the weight of each asset is:

Investment A weight = \$100,000 / \$400,000 = 25%

Investment B weight = \$300,000 / \$400,000 = 75%

Thus, the total expected return of the portfolio is:

Portfolio expected return = (25% x 5%) + (75% x 10%) = 8.75%

Portfolio variance is slightly more complicated; it is not a simple weighted average of the investments' variances. Because the two assets may move in relation to each other, their correlation must be taken into account. For this example, assume the correlation between the two investments is 0.65. Assume also the standard deviation (the square root of variance) for Investment A is 7% and the standard deviation for Investment B is 14% The portfolio variance for a two-asset portfolio is found using the following equation:

Portfolio variance = w(1) ^ 2 x o(1) ^ 2 + w(2) ^ 2 x o(2) ^ 2 + (2 x w(1) x w(2) x o(1) x o(2) x p)

Where,

w(1) = the portfolio weight of Investment A

o(1) = the standard deviation of Investment A

w(2) = the portfolio weight of Investment B

o(2) = the standard deviation of Investment B

p = the correlation between Investment A and Investment B

In this example, the portfolio variance is:

Portfolio variance = (25% ^ 2 x 7% ^ 2) + (75% ^ 2 x 14% ^ 2) + (2 x 25% x 75% x 7% x 14% x 0.65) = 0.0137

The portfolio standard deviation is the square root of this number, or 11.71%.